Related papers: Action-angle coordinates for the pendulum problem
In this note we deduce well known modular identities for Jacobi theta functions using the spectral representations associated with the real valued Brownian motion taking values on $[-1,+1]$. We consider two cases: (i) reflection at $-1$ and…
The small angle approximation often fails to explain experimental data, does not even predict if a plane pendulum's period increases or decreases with increasing amplitude. We make a perturbation ansatz for the Conserved Energy Surfaces of…
Previous works have considered the leading correction term to the scaled limit of various correlation functions and distributions for classical random matrix ensembles and their $\beta$ generalisations at the hard and soft edge. It has been…
The problem of the motion of a particle in an asymmetric double well is solved explicitly in terms of the Weierstrass and Jacobi elliptic functions. While the solution of the orbital motion is expressed simply in terms of the Weierstrass…
The closed form solution is found for the fully nonlinear dynamics of the gyroscope with a fixed point at the tip. The solution is found by using Cardano's formulae to factor a cubic, in the case where all roots are known to be real. From…
This report provides some closed form solutions -- and their inversion -- to a satellite's bounded motion on the equatorial plane of a spheroidal attractor (planet) considering the $J_{2}$ spherical zonal harmonic. The equatorial track of…
General analytical solutions of the Quantum Hamilton Jacobi Equation for conservative one-dimensional or reducible motion are presented and discussed. The quantum Hamilton's characteristic function and its derivative, i.e. the quantum…
It was shown that pedal coordinates provides natural framework in which to study force problems of classical mechanics in the plane. A trajectory of a test particle under the influence of central and Lorentz-like forces can be translated…
A trajectory isomorphism between the two Newtonian fixed center problem in the sphere and two associated planar two fixed center problems is constructed by performing two simultaneous gnomonic projections in $S^2$. This isomorphism converts…
Classical and quantum correlation functions are derived for a system of non-interacting particles moving on a circle. It is shown that the decaying behaviour of the classical expression for the correlation function can be recovered from the…
We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space, generalizing Lempert's formula for the convex case. This function is also known as the pluricomplex Green function…
In this paper we introduce new class of nonlinear interactions of $\zeta$-oscillating systems. The main formula is generated by corresponding subset of the set of trigonometric functions. Next, the main formula generates certain set of…
In this paper we obtain new canonical synergetic formula, namely an $\zeta$-analogue of next elementary trigonometric formula. This one describes cooperative interactions between corresponding class of elementary functions and the Riemann's…
A transformation is found between the one dimensional Schroedinger equation and a pendulum problem. It is demonstrated how to construct exact solutions with the resulted pendulum equation. The relation of this transformation to the…
The equations of motion for spinning compact binaries on eccentric orbits are treated perturbatively in powers of a fractional mass-difference ordering parameter. The solution is valid through first order in the mass-difference parameter. A…
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle and angular momentum is solved. While a key element for the construction of Wigner functions for the planar phase space…
Using elementary geometric tools, we apply essentially the same methods to derive expressions for the rotation angle of the swing plane of Foucault's pendulum and the rotation angle of the spin of a relativistic particle moving in a…
The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy $E$, the axial component of the angular momentum, $L_{z}$, and Carter's constant $Q$. These parameters are obtained by…
The relation that exists in quantum mechanics among action variables, angle variables and the phases of quantum states is clarified, by referring to the system of a generalized oscillator. As a by-product, quantum-mechanical meaning of the…
In this article we solve a class of two parameter polynomial-quintic equation. The solution follows if we consider the Jacobian elliptic function $sn$ and relate it with the coefficients of the equation. The solution is the elliptic…